Digital Object Identifier (DOI) 10.1007/s00205-013-0706-9Arch. Rational Mech. Anal.

The Nonlinear Heat Equation on W -RandomGraphs

Georgi S. Medvedev

Communicated by W. E

Abstract

For systems of coupled differential equations on a sequence of W -randomgraphs, we derive the continuum limit in the form of an evolution integral equation.We prove that solutions of the initial value problems (IVPs) for the discrete modelconverge to the solution of the IVP for its continuum limit. These results combinedwith the analysis of nonlocally coupled deterministic networks in Medvedev (Thenonlinear heat equation on dense graphs and graph limits. ArXiv e-prints, 2013)justify the continuum (thermodynamic) limit for a large class of coupled dynamicalsystems on convergent families of graphs.

1. Introduction

In this paper, we study coupled dynamical systems on a sequence of graphs{Gn}:

d

dtuni (t) = n−1

∑

j :(i, j)∈E(Gn)

D(unj − uni ), i ∈ [n] := {1, 2, . . . , n}, (1.1)

where Gn = 〈V (Gn), E(Gn)〉 is a graph on n nodes, and V (Gn) = [n] and E(Gn)

stand for the sets of nodes and edges of Gn respectively. D is a Lipschitz continuousfunction. The operator on the right-hand side of (1.1) models the nonlinear diffusionacross edges of Gn . Thus, we refer to (1.1) as a nonlinear heat equation on Gn .

The evolution equations like (1.1) are used in modeling diverse systems rangingfrom neuronal networks in biology [7,9,22,32], to Josephson junctions and coupledlasers in physics [16,30], to communication, sensor, and power networks in tech-nology [6,19]. The Kuramoto model, a prominent example of (1.1), is widely usedas a paradigm for studying collective dynamics of coupled oscillators of diversenature [6,8,11,12,35].

Georgi S. Medvedev

In this paper, we are interested in the case when {Gn} is a sequence of densegraphs, that is, |E(Gn)| = O(n2). This corresponds to the nonlocal diffusionoperator in (1.1). Nonlocally coupled systems have attracted much attention innonlinear science recently [8,14,29,31,33–35]. They arise as models of diversephenomena throughout physics and biology and feature several remarkable effects,such as chimera states and coherence-incoherence transition (see, for example,[13–15,25–28,31]). Overall, nonlocally coupled dynamical systems are less under-stood than systems with local coupling.

For analyzing nonlocally coupled systems, the continuum (thermodynamic)limit proved to be a very useful tool [8,14,29,35]. As n → ∞, one can formallyinterpret the right-hand side of (1.1) as a Riemann sum to obtain

∂

∂tu(x, t) =

∫

IW (x, y)D(u(y, t) − u(x, t)) dy, (1.2)

where u(x, t) now describes a continuum of (local) dynamical systems distributedalong I := [0, 1]. For some patterns of connectivity, the kernel W in (1.2) can beguessed from the pixel picture of the adjacency matrix of Gn [4,17]. For example, letGn be a graph on n nodes distributed uniformly along a circle, and let k = �rn� forfixed r ∈ (0, 1). Suppose each node of Gn is connected to k of its nearest neighborsfrom each side, that is, Gn is a k-nearest-neighbor graph. The pixel picture of Gn

is shown in Fig. 1a. Specifically, Fig. 1a shows the support of the {0, 1}-valuedfunction WGn : I 2 → {0, 1} such that

WGn (x, y) = 1 if (i, j) ∈ E(Gn) and (x, y) ∈ [(i − 1)n−1, in−1)

×[( j − 1)n−1, jn−1), (i, j) ∈ [n]2.

Function WGn provides the geometric representation of the adjacency matrix ofGn . It is easy to see that as n → ∞, {WGn } converges to the {0, 1}-valued function,whose support is shown in Fig. 1b. This is the limit of the k-nearest-neighbor familyof graphs {Gn}.

a

0 0.25 0.5 0.75 11

0.75

0.5

0.25

0b

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

Fig. 1. The plot of the support of WGn (a) and that of the support of its limit WG∞ (b). Eachfunction is defined on a unit square and is equal to 1 on the colored regions and 0 otherwise.The direction of the vertical axis was chosen to emphasize the relation of WGn (a) to theadjacency matrix of the corresponding graph

The Nonlinear Heat Equation on W -Random Graphs

The formally derived continuum limit (1.2) was used to study the discrete model(1.1) for large n in many papers [8,14,29,34,35]. In [21], we provided a rigorousjustification of the continuum limit (1.2). The analysis of the continuum limit in [21]uses the ideas from the theory of graph limits [5,17,18], which for every convergentfamily of dense graphs defines the limiting object, a measurable symmetric functionW . This function is called a graphon. It captures the connectivity of Gn for large n.In [21], for convergent sequences of deterministic graphs {Gn}, it was shown thatwith the kernel of the integral operator on the right-hand side of (1.2) taken to bethe limit of {Gn}, the solution of the IVP for (1.2) approximates those of the IVPsfor (1.1) for large n.

The analysis in [21] does not cover dynamical systems on random graphs.The latter have many important applications [34,35]. Thus, in this paper, we focuson systems on random graphs. Specifically, we prove convergence of solutions ofthe IVPs for (1.1) on W -random graphs Gn to the solution of the IVP for (1.2).A W -random graph is constructed from a graphon W [17,18]. This constructionprovides a convenient general analytical model for random graphs, which includesmany random graphs that are important in applications, such as Erdos–Rényi andsmall-world (SW) graphs (see Figs. 1c and 2b, c) [3,10,34]. At the same time, W -random graphs fit naturally into the convergence analysis of the families of discretemodels like (1.1).

The remainder of this paper is organized as follows. In the next section, weformulate the IVPs for the discrete model and its continuum limit. In Sections 3and 4, we prove convergence of solutions of discrete models for two differentvariants of W -random graphs. In the variant, analyzed in Section 3, the right-handside of (1.1) can be interpreted as the Monte-Carlo approximation of the integralon the right-hand side of (1.2). Consistent with this interpretation, we find thatthe rate of convergence of the solutions of discrete problems (in C(0, T ; L2(I ))norm) is O(n−1/2). In the variant of the random network model considered inSection 4, which was included for the sake of convenience in applications, the rateof convergence also depends on the regularity of the graphon W . As an applicationof our results, in Section 5 we derive the continuum limit for dynamical systemson SW graphs [34,35] (see Fig. 2). We conclude with the discussion of our resultsin Section 6.

a b c

Fig. 2. The pixel pictures of the k-nearest-neighbor network on a ring (a) and two small-world graphs (b, c) that were obtained from the network in (a) by replacing local connectionswith random long-range connections

Georgi S. Medvedev

2. The Discrete Model and Its Continuum Limit

Throughout this paper, we assume that W (x, y) belongs to W0, a class ofsymmetric measurable functions on I 2 with values in I . W represents the limit ofa convergent family of dense graphs {Gn} (see [17], for an exposition of the theoryof graph limits; see also Section 2 in [21] for a brief review of facts from this theorythat are relevant for constructing continuum limits of dynamical networks.)

Let Xn = {xn1, xn2, . . . , xnn} be a set of distinct points from I and W ∈W0. In this section, we introduce IVPs for the nonlinear heat equation on Gn =〈V (Gn), E(Gn)〉, a certain graph on n nodes, constructed using W and Xn .

The sequence of graphs {Gn} will be defined below. Suppose Gn is given. Bythe IVP for the nonlinear heat equation on Gn , we mean

d

dtuni (t) = n−1

∑

j :(i, j)∈E(Gn)

D(unj − uni ), (2.1)

uni (0) = g(xi ), i ∈ [n], (2.2)

where un(t) = (un1(t), un2(t), . . . , unn(t)) is the unknown function. Here, D(·) isa Lipschitz function on R and g is a bounded measurable function on I .

The solution of the IVP for the discrete model (2.1), (2.2) will be comparedwith the solution of the IVP for the continuum limit

∂

∂tu(x, t) =

∫

IW (x, y)D (u(y, t) − u(x, t)) dy, (2.3)

u(x, 0) = g(x), x ∈ I. (2.4)

For W ∈ W0, g ∈ L∞(I ), and a Lipschitz continuous D, there is a unique strongsolution of (2.3), (2.4) u ∈ C1(R; L∞(I )) [21]. Here and below, we use bold fontto denote vector-valued functions, for example, u(t) = u(·, t) ∈ L∞(I ).

Denote the projection of the solution of the continuous problem (2.3), (2.4),u(x, t) onto Xn by

PXn u(x, t) = (u(xn1, t), u(xn2, t), . . . , un(xnn, t)).

Both functions un(t) and PXn u(x, t) are defined on the discrete set Xn . Forsuch functions, we will use the weighted Euclidean inner product

(u, v)n = 1

n

n∑

i=1

uivi , u = (u1, u2, . . . , un)ᵀ, v = (v1, v2, . . . , vn)ᵀ

and the corresponding norm ‖u‖2,n = √(u, u)n . Below, we will use ‖ · ‖2,n to

study the difference between the solutions of the discrete and continuous problems(2.1) and (2.3) on W -random graphs.

The Nonlinear Heat Equation on W -Random Graphs

3. Networks on W -Random Graphs Generated by Random Sequences

Denote

X = (x1, x2, x3, . . .) and Xn = (x1, x2, . . . , xn), (3.1)

where xi , i ∈ N are independent identically distributed (IID) random variables(RVs). RV x1 has uniform on I distribution, that is, L(x1) = U (I ).

Definition 3.1. (cf. [18]) By a W -random graph on n nodes generated by the randomsequence X , denoted Gn = G(Xn, W ), we mean Gn = 〈[n], E(Gn)〉 such that theedges of Gn are selected at random and

P{(i, j) ∈ E(Gn)} = W (xi , x j ), for each (i, j) ∈ [n]2, i �= j.

The decision whether to include a pair (i, j) ∈ [n]2, i �= j, is made independentlyfrom the decisions for other pairs.

Remark 3.2. The graph sequence {Gn} converges to graphon W almost surely asn → ∞ [18].

Theorem 3.3. Suppose W ∈ W0, D is a Lipschitz continuous function on R, andg ∈ L∞(I ). Let T > 0 and suppose that the solution of the IVP (2.3) and (2.4)u(x, t) satisfies the following inequality

mint∈[0,T ]

∫

I

{∫

IW (x, y)D (u(y, t) − u(x, t))2 dy

−(∫

IW (x, y)D (u(y, t) − u(x, t)) dy

)2}

dx � C1 (3.2)

for some positive constant C1. Then the solutions of the IVPs for the discrete andcontinuum models (2.1), (2.2) and (2.3), (2.4) satisfy the following relation

limn→∞ P

{n1/2 sup

t∈[0,T ]‖u(n)(t) − PXn

u(x, t)‖2,n � C

}= 1

for some constant C > 0.

Remark 3.4. The integral expression in (3.2) defines a continuous function of t .This follows from u ∈ C(0, T ; L∞(I )), ‖W‖L∞(I 2)=1, and Lipschitz continuity ofD. This justifies the use of min in (3.2).

For the proof of this theorem we will need the following application of theCentral Limit Theorem (CLT) [2].

Lemma 3.5. Suppose W ∈ W0, f ∈ L∞(I 2), and

X = (x1, x2, x3, . . .),

Georgi S. Medvedev

where xi , i ∈ N, are IID RVs with L(x1) = U (I ). Define RVs {ξi j }, (i, j) ∈ N2,

such that L(ξi j |X) = Bin(W (xi , x j )).1 Specifically,

P(ξi j = 1|X) = W (xi , x j ) and P(ξi j = 0|X) = 1 − W (xi , x j ). (3.3)

Further, let

ηi j = ξi j f (xi , x j ), (i, j) ∈ N2, (3.4)

zni = 1

n

n∑

j=1

ηi j −∫

If (xi , y)W (xi , y) dy, and Sn =

n∑

i=1

z2ni . (3.5)

Finally, we assume

σ 2 :=∫

I 2f (x, y)2W (x, y) dx dy −

∫

I

(∫

If (x, y)W (x, y) dy

)2

dx > 0.

(3.6)

Then

Sn − σ 2

n−1/2√

5σ 4 + O(n−1)

d−→ N (0, 1), (3.7)

whered−→ denotes convergence in distribution, and N (0, 1) stands for the standard

normal distribution.

By construction, {ηi j } are IID RVs. Moreover, from (3.3) and (3.4) we have

μ(xi ) = E(ηi j |xi ) =∫

If (xi , y)W (xi , y) dy. (3.8)

Therefore,

μ := Eηi j = EE(ηi j |xi ) =∫

I 2f (x, y)W (x, y) dx dy, (3.9)

Vηi j = EE((ηi j − μ)2|xi ) = EE((η2i j |xi ) − 2μE(ηi j |xi ) + μ2)

=∫

I 2f (x, y)2W (x, y) dx dy −

∫

I

(∫

If (x, y)W (x, y) dy

)2

dx = σ 2.

(3.10)

Let

yni = √nzni . (3.11)

1 Bin(p) stands for the binomial distribution with parameter p ∈ [0, 1].

The Nonlinear Heat Equation on W -Random Graphs

We prove (3.7) by applying the CLT to∑n

i=1 y2ni . To justify the application of the

CLT, we need to compute three first moments of y2ni . To this end,

Ey2ni = n−1

E E

⎛

⎝∑

1� j,k�n

(ηi j − μ(xi ))(ηik − μ(xi ))|xi

⎞

⎠

= E E

⎛

⎝∑

1� j�n

(ηi j − μ(xi ))2|xi

⎞

⎠

+2n−1E E

⎛

⎝∑

1� j<k�n

(ηi j − μ(xi ))(ηik − μ(xi ))|xi

⎞

⎠ . (3.12)

The first term on the right hand side of (3.12) is equal to σ 2 [see (3.10)]. Thesecond term is equal to 0, as easy to see using the independence of ηi j −μ(xi ) andηik − μ(xi ) for k �= j . Thus,

Ey2ni = σ 2 + 2n−1

E

⎛

⎝∑

1� j<k�n

E(ηi j − μ(xi )|xi )E(ηik − μ(xi )|xi )

⎞

⎠ = σ 2.

(3.13)

Recall that σ 2 > 0, by (3.6). Similarly, we compute

E(y4ni ) = n−2

E E

⎛

⎝∑

1� j1, j2, j3, j4�n

(ηi j1 − μ(xi )) · · · (ηi j4 − μ(xi ))|xi

⎞

⎠

= 6n−2E

⎛

⎝∑

1� j<k�n

E(ηi j − μ(xi )|xi )2 E(ηik − μ(xi )|xi )

2

⎞

⎠

+n−2E

⎛

⎝∑

1� j�n

E(ηi j − μ(xi )|xi )4

⎞

⎠

= 6n(n − 1)

n2 σ 4 + O(n−1) = 6σ 4 + O(n−1) (3.14)

and

Ey6ni = n−3

E E

⎛

⎝∑

1� j1, j2, j3, j4, j5, j6�n

(ηi j1 − μ(xi )) · · · (ηi j6 − μ(xi ))|xi

⎞

⎠

Georgi S. Medvedev

=(

6

2

)(4

2

)n−3

E

⎛

⎝∑

1� j<k<l�n

E(ηi j − μ(xi )|xi )2 E(ηik

−μ(xi )|xi )2 E(ηil − μ(xi )|xi )

2

⎞

⎠ + O(n−1)

= 90n(n − 1)(n − 2)

n3 σ 6 + O(n−1) = 90σ 6 + O(n−1). (3.15)

For n ∈ N, let

ζni := y2ni − Ey2

ni√nV(y2

in)

= y2ni − σ 2

n1/2√

5σ 4 + O(n−1), i ∈ [n], (3.16)

where (3.13) and (3.14) were used to obtain the expression on the right hand side.Consider

ζn1, ζn2, . . . , ζnn . (3.17)

By construction, ζni , i ∈ [n], are IID RVs. Further,

Eζni = 0 and V

(n∑

i=1

ζni

)= 1. (3.18)

Moreover, the triangular array (3.17) satisfies the Lyapunov condition [2]n∑

i=1

E|ζni |3 �∑n

i=1 E(y6ni + 3y4

niσ2 + 3y2

niσ4 + σ 6)

n3/2(5σ 4 + O(n−1))3/2

= O(n−1/2) → 0 as n → ∞. (3.19)

From (3.18) and (3.19), via the CLT, we conclude that∑n

i=1(y2ni − σ 2)

√n(5σ 4 + O(n−1))

= n−1 ∑ni=1 y2

ni − σ 2

n−1/2√

5σ 4 + O(n−1)

d−→ N (0, 1) n → ∞. (3.20)

The statement (3.7) follows from (3.20) and the definition of yni (3.11). � For the proof of Theorem 3.3, we need to extend Lemma 3.5 to cover the case

when f depends on t ∈ [0, T ] in addition to (x, y) ∈ I 2.

Corollary 3.6. Suppose that f in Lemma 3.5 also depends on t ∈ [0, T ], andf ∈ C(0, T ; L∞(I 2)) if viewed as a mapping from [0, T ] to L∞(I 2), f(t) =f (·, t) ∈ L∞(I 2). Adding t−dependence to all variables defined using f and,otherwise, keeping the notation of Lemma 3.5, we assume that

mint∈[0,T ] σ

2(t) � c1 > 0. (3.21)

Then the conclusion of Lemma 3.5 holds for t−dependent sums for every t ∈ [0, T ]Sn(t) − σ 2

n (t)

n−1/2√

5σ 4n (t) + O(n−1)

d−→ N (0, 1) as n → ∞. (3.22)

The Nonlinear Heat Equation on W -Random Graphs

Proof. From the assumption f ∈ C(0, T ; L∞(I 2)) and (3.21), for

σ 2(t) =∫

I 2f (x, y, t)2W (x, y) dx dy −

∫

I

(∫

If (x, y, t)W (x, y) dy

)2

dx,

we have

0 < c1 � σ 2(t) � 2‖f‖2C(0,T ;L∞(I 2))

. (3.23)

With these bounds, by repeating the steps in the proof of Lemma 3.5, we first showthat t−dependent moments of y2

ni (t) are bounded uniformly in t ∈ [0, T ]; thenverify Lyapunov condition for every t ∈ [0, T ] and apply the CLT. This shows(3.22). �

We are now in a position to prove Theorem 3.3.

Proof of Theorem 3.3. Denote ζni (t) = u(xi , t) − uni (t), i ∈ [n] and let

ζn(t) = (ζn1(t), ζn2(t), . . . , ζnn(t)).

By subtracting Equation i in (2.1) from the corresponding equation in (2.3) evalu-ated at x = xi , we have

d

dtζni (t) = zni (t) + 1

n

n∑

j=1

ξi j [D(u(x j , t) − u(xi , t)) − D(unj (t) − uni (t))],

(3.24)

where

zni =∫

IW (xi , y)D(u(y, t) − u(xi , t)) dy − 1

n

n∑

j=1

ξi j D(u(x j , t) − u(xi , t)),

(3.25)

and ξi j are defined in (3.3).Next, we multiply both sides of (3.24) by n−1ζni and sum over i to obtain

1

2

d

dt‖ζn‖2

2,n = (zn, ζn)n + 1

n2

n∑

i, j=1

ξi j [D(u(x j , t)

−u(xi , t)) − D(unj (t) − uni (t))]ζni , (3.26)

where zn = (zn1, zn2, . . . , znn). We estimate the first term on the right-hand sideof (3.26) via the Cauchy–Schwarz inequality

|(zn, ζn)n| � ‖zn‖2,n‖ζn‖2,n � 2−1(‖zn‖22,n + ‖ζn‖2

2,n). (3.27)

Georgi S. Medvedev

For the second term we use the Lipschitz continuity of D, |ξi j | � 1, the Cauchy–Schwarz inequality, and the triangle inequality to obtain

∣∣∣∣∣∣1

n2

n∑

i, j=1

ξi j [D(u(x j , t) − u(xi , t)) − D(unj (t) − uni (t))]ζni

∣∣∣∣∣∣

� L

n2

n∑

i, j=1

(|ζnj (t)| + |ζni (t)|)|ζni (t)| � 2L‖ζn(t)‖22,n . (3.28)

Using (3.26), (3.27), and (3.28), we have

d

dt‖ζn‖2

2,n � (4L + 1)‖ζn‖22,n + ‖zn‖2

n,2. (3.29)

From (3.29) via the Gronwall’s inequality we have

supt∈[0,T ]

‖ζn(t)‖2,n �supt∈[0,T ] ‖zn(t)‖2

2,n

4L + 1exp{(4L + 1)T }. (3.30)

It remains to bound supt∈[0,T ] ‖zn(t)‖22,n . To this end, let

f (x, y, t) := D(u(y, t) − u(x, t)).

Using u ∈ C(0, T ; L∞(I )), Lipschitz continuity of D, and the triangle inequality,we have

‖f‖C(0,T ;L∞(I 2)) � L maxt∈[0,T ] ess sup(x,y)∈I 2 |u(x, t) − u(y, t)|

� 2L‖u‖C(0,T ;L∞(I )). (3.31)

By (3.2) and (3.31), we find that σ 2(t) is bounded for t ∈ [0, T ]C1 � σ 2(t) � 2L‖u‖C(0,T ;L∞(I )) =: C2. (3.32)

Using Corollary 3.6, for zn = (zn1, zn2, . . . , znn) [see (3.25)], we have

n‖zn‖22,n(t) − σ 2(t)

n−1/2β(σ 2(t))d→ N (0, 1), where β(σ 2(t)) =

√5σ 2(t) + O(n−1).

Further, we have

P(|n‖zn(t)‖22,n − σ 2(t)| > 1) = P

( |n‖zn(t)‖22,n − σ 2(t)|

n−1/2β(σ 2(t))>

n1/2

β(σ 2(t))

)

� P

( |n‖zn(t)‖22,n − σ 2(t)|

n−1/2β(σ 2(t))>

n1/2

C2

)→ 0,

(3.33)

The Nonlinear Heat Equation on W -Random Graphs

as n → ∞. We used (3.32) to obtain the last inequality in (3.33). Convergencein (3.33) is uniform for t ∈ [0, T ]. Therefore, ‖zn(t)‖2

n,2 converges to zero inprobability uniformly in t . Moreover,

P(‖zn(t)‖2n,2 > (C2 + 1)n−1) � P(|n‖zn(t)‖2

2,n − σ 2(t)| > 1)→0, as n → ∞

uniformly for t ∈ [0, T ].Let ε > 0 be arbitrary. Then for C3 := C2 + 1 and for some N ∈ N, we have

P

(sup

t∈[0,T ]‖zn(t)‖2

2,n > C3n−1

)< ε for n > N .

The combination of this and (3.30) proves the theorem. �

4. Networks on W -Random Graphs Generated by Deterministic Sequences

In this section, we consider the heat equations on W -random graphs generatedby deterministic sequences of points from I . To this end, we partition I into nsubintervals

Ini = [(i − 1)n−1, in−1), i ∈ [n − 1], and Inn = [(n − 1)n−1, 1]. (4.1)

Suppose

Xn = {xn1, xn2, . . . , xnn}, xni ∈ Ini i ∈ [n], (4.2)

where Ini denotes the closure of Ini .

Definition 4.1. Graph Gn = 〈V (Gn), E(Gn)〉 is called a W -random graph gen-erated by the deterministic sequence Xn and is denoted Gn = G(W, Xn), ifV (Gn) = [n] and for every (i, j) ∈ [n]2, i �= j,

P{(i, j) ∈ E(Gn)} = W (xni , xnj ).

The decision whether to include (i, j) to E(Gn) is made independently for eachpair (i, j) ∈ [n]2, i �= j .

Remark 4.2. If W is continuous on I 2 almost everywhere, then {G(W, Xn)} isconvergent with the limit given by graphon W (cf. Lemma 2.5 [4]).

Let un(t) = (un1(t), un2(t), . . . , unn(t)) denote the solution of the IVP (2.1),(2.2) for the heat equation on Gn = G(W, Xn), and define un : I × R → R asfollows. For x ∈ Ini , i ∈ [n], let

un(x, t) = uni (t), t ∈ R.

Georgi S. Medvedev

Theorem 4.3. Suppose W ∈ W0 is almost everywhere continuous on I 2, D : R →R is Lipschitz continuous, and g ∈ L∞(I ). Let u(x, t) denote the solution of theIVP (2.3), (2.4). Suppose further

mint∈[0,T ]

∫

I 2D(u(y, t) − u(x, t))W (x, y)(1 − W (x, y)) dx dy > 0 (4.3)

for some T > 0.2 Then

‖un − u‖C(0,T ;L2(I ))p→ 0 as n → ∞. (4.4)

The convergence in (4.4) is in probability.

For the proof of Theorem 4.3 we need to derive several auxiliary results. Thefirst result is parallel to Lemma 3.5 of the previous section.

Lemma 4.4. Let {Wni j } and { fni j } be two real arrays defined for n ∈ N andi, j ∈ [n], and

σ 2ni = n−1

n∑

i=1

f 2ni j Wni j (1 − Wni j ), i ∈ [n], (4.5)

σ 2n = n−1

n∑

i=1

σ 2ni . (4.6)

Assume that { fni j }, n ∈ N, i, j ∈ [n], is a bounded array, 0 � wni j � 1 and

lim infn→∞ σ 2

n > 0. (4.7)

Let {ξni j }, n ∈ N, (i, j) ∈ [n]2 be independent binomial RVs L(ξni j ) =Bin(Wni j ). Further, let

ηni j = ξni j fni j , (i, j) ∈ [n]2,

zni = 1

n

n∑

j=1

(ηni j − fni j Wni j ),

Sn =n∑

i=1

z2ni .

Then

Sn − σ 2n

n−1/2√

5σ 4n + O(n−1)

d−→ N (0, 1) as n → ∞. (4.8)

2 Because u ∈ C(R, L∞(I )), D is Lipschitz, and W is bounded, the integral in (4.3)defines a continuous function of t . Thus, the use min in (4.3) is justified.

The Nonlinear Heat Equation on W -Random Graphs

Proof. First, compute the moments of the independent RVs {ηni j }, n ∈ N, (i, j) ∈[n]2,

Eηkni j = f k

ni j Wni j , k ∈ N.

Thus, for yni = √nzni , i ∈ [n], we have Eyni = 0. Further,

Ey2ni = n−1

E

⎛

⎝∑

1� j,k�n

(ηni j − fni j Wni j )(ηnik − fni j Wni j )

⎞

⎠

= n−1E

⎛

⎝∑

1� j�n

(ηni j − fni j Wni j )2

⎞

⎠

+2n−1E

⎛

⎝∑

1� j<k�n

(ηni j − fni j Wni j )(ηik − fnik Wnik)

⎞

⎠

= σ 2ni + 2n−1

∑

1� j<k�n

E(ηni j − fni j Wni j )E(ηnik − fnik Wnik) = σ 2ni ,

where

σ 2ni := n−1

E

⎛

⎝∑

1� j�n

(ηni j − fni j Wni j )2

⎞

⎠=n−1n∑

j=1

f 2ni j Wni j (1 − Wni j ). (4.9)

Similarly, we compute

Ey4ni = n−2

E

⎛

⎝∑

1� j1, j2, j3, j4�n

(ηni j1 − fni j1 Wni j1) · · · (ηni j4 − fni j4 Wni j4)

⎞

⎠

= 6n−2∑

1� j<k�n

E(ηni j − fni j Wni j )2 E(ηnik − fnik Wnik)

2

+ n−2∑

1� j�n

E(ηni j − fni j Wni j )4

= 6n(n − 1)

n2 σ 4ni + O(n−1) = 6σ 4

ni + O(n−1).

and

Ey6ni = n−3

E

⎛

⎝∑

1� j1, j2, j3, j4, j5, j6�n

(ηni j1 − fni j1 Wni j1) · · · (ηni j6 − fni j6 Wni j6)

⎞

⎠

=(

6

2

)(4

2

)n−2

∑

1� j<k<l�n

E(ηni j − fni j Wni j )2

× E(ηnik − fnik Wnik)2E(ηnil − fnil Wnil)

2 + O(n−1)

= 90n(n − 1)(n − 2)

n3 σ 6ni + O(n−1) = 90σ 6

ni + O(n−1).

Georgi S. Medvedev

For n ∈ N, let

ζni := y2ni − Ey2

ni√nV(y2

in)

= y2ni − σ 2

ni

n1/2√

5σ 4ni + O(n−1)

, i ∈ [n]. (4.10)

Consider

ζn1, ζn2, . . . , ζnn . (4.11)

By construction, ζni , i ∈ [n] are independent RVs. Further,

Eζni = 0 and V

(n∑

i=1

ζni

)= 1. (4.12)

Moreover, the triangular array (3.17) satisfies the Lyapunov condition [2]

n∑

i=1

E|ζni |3 �∑n

i=1 E(y6ni + 3y4

niσ2ni + 3y2

niσ4ni + σ 6

ni )

n3/2(5σ 4n + O(n−1))3/2

= O(n−1/2) → 0 as n → ∞. (4.13)

From (4.12) and (4.13), using the CLT, we conclude that

∑ni=1(y2

ni − σ 2ni )√

n(5σ 4n + O(n−1))

= n−1 ∑ni=1 y2

ni − σ 2n

n−1/2√

5σ 4n + O(n−1)

d−→ N (0, 1), n → ∞. (4.14)

The statement (4.8) follows from (4.14) and the definition of yni . � With obvious modifications the proof of Lemma 4.4 can be easily extended to

cover the following version of the lemma.

Corollary 4.5. Suppose fni j in Lemma 4.4 depend on real parameter t ∈ [0, T ]for some T . Keeping the notation of Lemma 4.4, we add t−dependence to allvariables defined using fni j (t). Assume that functions fni j (t), n ∈ N, i, j,∈ [n],are uniformly bounded for t ∈ [0, T ] and

lim infn→∞ σ 2

n (t) = lim infn→∞ n−1

n∑

i, j=1

fni j (t)Wni j (1 − Wni j ) � C1 > 0 (4.15)

for every t ∈ [0, T ].Then the conclusion of Lemma 4.4 holds for t−dependent sums for every t ∈

[0, T ]Sn(t) − σ 2

n (t)

n−1/2√

5σ 4n (t) + O(n−1)

d−→ N (0, 1), n → ∞.

The Nonlinear Heat Equation on W -Random Graphs

Having prepared the application of the CLT that will be needed in the proof ofTheorem 4.3, we now introduce an auxiliary IVP for the heat equation on a weightedgraph Gn = H(W, Xn). The latter is a complete graph on n nodes, V (Gn) = [n].Each edge of Gn is supplied with the weight

Wni j = W (xni , xnj ), (i, j) ∈ [n]2, i �= j.

Consider the IVP for the heat equation on the weighted graph Gn

d

dtvni (t) = n−1

∑

j :(i, j)∈E(Gn)

Wni j D(vnj − vni ), (4.16)

vni (0) = g(xi ), i ∈ [n]. (4.17)

Denote the solution of the IVP (4.16) and (4.17) by vn(t) = (vn1(t), vn2(t), . . . ,vnn(t)). Let vn(x, t) be a function defined on I ×R and such that for x ∈ Ini , i ∈ [n]

vn(x, t) = vn(t), t ∈ R.

Next, define a step-function Wn on I 2 such that for (x, y) ∈ Ini × Inj , i, j ∈ [n],Wn(x, y) = Wni j .

By construction, vn(x, t) solves the following IVP

∂

∂tvn(x, t) =

∫

IWn(x, y)D(vn(y, t) − vn(x, t)) dy, (4.18)

vn(x, 0) = g(xni ), x ∈ Ini , i ∈ [n]. (4.19)

It was shown in [21] that for large n, vn(x, t) approximates the solution of the IVP(2.3), (2.4). Specifically, we have the following lemma.

Lemma 4.6. [21, Theorem 5.2] Suppose W ∈ L∞(I 2) is almost everywhere con-tinuous on I 2, D is Lipschitz continuous, and g ∈ L∞(I ). Then for any T > 0

‖u − vn‖C(0,T ;L2(I )) → 0 as n → ∞. (4.20)

We use Lemma 4.6 to derive the following result.

Lemma 4.7. Suppose W ∈ W0 is almost everywhere continuous on I 2, D is Lip-schitz continuous, and g ∈ L∞(I ). Let u(x, t) and vn(x, t) denote the solutions ofthe IVPs (2.3), (2.4) and (4.18), (4.19), respectively; and let

σ 2(t) =∫

I 2D(u(y, t) − u(x, t))W (x, y)(1 − W (x, y)) dx dy,

σ 2n (t) =

∫

I 2D(vn(y, t) − vn(x, t))Wn(x, y)(1 − Wn(x, y)) dx dy.

Then

supt∈[0,T ]

|σ 2n (t) − σ 2(t)| � C2[‖vn − u‖C(0,T ;L2(I )) + ‖Wn − W‖L2(I 2)],

for some C2 > 0. In particular, σ 2n → σ 2 uniformly for t ∈ [0, T ].

Georgi S. Medvedev

Proof. 1. Using Lipschitz continuity of D and the triangle inequality, for anyt ∈ [0, T ] we have

∣∣∣∣∫

I 2D(vn(y, t) − vn(x, t)) − D(u(y, t) − u(x, t)) dx dy

∣∣∣∣

� L∫

I 2|vn(y, t) − u(y, t)| + |vn(x, t) − u(x, t)| dx dy

� 2L‖vn − u‖C(0,T ;L2(I )) → 0, (4.21)

as n → ∞. Therefore,

maxt∈[0,T ]

∣∣∣∣∫

I 2D(vn(y, t) − vn(x, t)) dx dy

∣∣∣∣ � C3, n ∈ N, (4.22)

for some C3 independent of n.2. Denote q(x) = x(1 − x). For x, y ∈ [0, 1], |q(x) − q(y)| � |x − y|. Thus,

|q(W ) − q(Wn)| � |W − Wn|. (4.23)

3. Finally, we estimate |σn(t) − σ(t)|. For arbitrary t ∈ [0, T ], we have∣∣∣∣∫

I 2D(vn(y, t) − vn(x, t))q(Wn(x, y)) dx dy

−∫

I 2D(u(y, t) − u(x, t))q(W (x, y)) dx dy

∣∣∣∣

�∣∣∣∣∫

I 2D(vn(y, t) − vn(x, t)) [q(Wn(x, y)) − q(W (x, y))] dx dy

∣∣∣∣

+∣∣∣∣∫

I 2[D(vn(y, t) − vn(x, t)) − D(u(y, t) − u(x, t))] q(W (x, y)) dx dy

∣∣∣∣ .

(4.24)

Using the Cauchy–Schwarz inequality, Lipschitz continuity of D, |q(W )| � 1,(4.22), and (4.23) from (4.24) we obtain

supt∈[0,T ]

|σn(t) − σ(t)| � C3‖W − Wn‖L2(I 2) + L‖vn − u‖C(0,T ;L2(I )).(4.25)

Note that Wn → W as n → ∞ at every point of continuity of W , that is, almosteverywhere on I 2. Therefore, by the dominated convergence theorem,

‖W − Wn‖L2(I 2) → 0 as n → ∞. (4.26)

The statement of the lemma follows from (4.25), (4.26), and Lemma 4.6.�

Proof of Theorem 4.3. Denote ηni (t) = uni (t) − vni (t), i ∈ [n], and

ηn(t) = (ηn1(t), ηn2(t), . . . , ηnn(t)).

The Nonlinear Heat Equation on W -Random Graphs

By subtracting Equation i in (4.16) from the corresponding equation in (2.1) writtenfor Gn = G(W, Xn), we have

d

dtηni = 1

n

⎛

⎝n∑

j=1

ξni j D(unj − uni ) −n∑

j=1

Wni j D(vnj − vni )

⎞

⎠ . (4.27)

By rewriting the right-hand side of (4.27), we obtain

d

dtηni = 1

n

n∑

j=1

ξni j [D(unj − uni ) − D(vnj − vni )] + zni , (4.28)

where

zni = 1

n

n∑

j=1

ξni j D(vnj − vni ) − 1

n

n∑

j=1

wni j D(vnj − vni ). (4.29)

By multiplying both sides of (4.28) by n−1ηni and summing over i , we have

1

2

d

dt‖ηn‖2

2,n = 1

n2

n∑

i, j=1

ξi j [D(unj − uni ) − D(vnj − vni )]ηni + (zn, ηn)n . (4.30)

We bound the first term on the right hand side of (4.30) using the Lipschitz continuityof D, |ξi j | � 1, the Cauchy–Schwarz inequality, and the triangle inequality

∣∣∣∣∣∣1

n2

n∑

i, j=1

ξi j [D(unj − uni ) − D(vnj − vni )]ηni

∣∣∣∣∣∣

� L

n2

n∑

i, j=1

(|ηnj | + |ηni |)|ηni | � 2L‖ηni‖22,n . (4.31)

We bound the second term using the Cauchy–Schwarz inequality

|(zn, ηn)n| � ‖zn‖2,n‖ηn‖2,n � 1

2(‖zn‖2

2,n + ‖ηn‖22,n), (4.32)

where zn = (zn1, zn2, . . . , znn).

The combination of (4.30), (4.31), and (4.32) yields

d

dt‖ηn‖2

2,n � (4L + 1)‖ηn‖22,n + ‖zn‖2

2,n . (4.33)

By Gronwall’s inequality,

maxt∈[0,T ] ‖η‖2

2,n �maxt∈[0,T ] ‖zn(t)‖2

2,n

4L + 1exp{(4L + 1)T }. (4.34)

Thus,

maxt∈[0,T ] ‖η‖2,n � maxt∈[0,T ] ‖zn(t)‖2,n√

4L + 1exp{(2L + 1)T }. (4.35)

Georgi S. Medvedev

It remains to estimate ‖zn(t)‖2,n [see (4.29)]. To this end, we use Corollary 4.5with

fni j (t) = D(vnj (t) − vni (t)) and Wni j = W (xni , xnj ).

From Lemma 4.7 and (4.3), we have

mint∈[0,T ] σ

2n (t) � C4 > 0, (4.36)

for sufficiently large n. In particular, (4.15) holds. Similarly, by Lemma 4.7, wehave

maxt∈[0,T ] σ

2n (t) � C5, n ∈ N. (4.37)

By Corollary 4.5, for arbitrary t ∈ [0, T ], we have

P{|n‖zn(t)‖22,n − σ 2

n (t)| > 1} = P

{∣∣∣∣∣n‖zn(t)‖2

2,n − σ 2n (t)

n−1/2√

5σ 4n (t) + O(n−1)

∣∣∣∣∣ >n1/2

√5σ 4

n (t) + O(n−1)

}

� P

⎧⎨

⎩

∣∣∣∣∣n‖zn(t)‖2

2,n − σ 2n (t)

n−1/2√

5σ 4n (t) + O(n−1)

∣∣∣∣∣ >n1/2

√5C2

5 + O(n−1)

⎫⎬

⎭ → 0 as n → ∞. (4.38)

Using (4.37), from (4.38) we have

P{‖zn(t)‖22,n � (C5 + 1)n−1} � P{|n‖zn(t)‖2

2,n − σ 2n (t)| > 1}→0 as n →∞.

(4.39)

Finally, since t ∈ [0, T ] is arbitrary from (4.39) we further have

limn→∞ P{ max

t∈[0,T ] ‖zn(t)‖2,n � C6n−1/2} = 0. (4.40)

The combination of (4.34) and (4.40) yields that ‖ηn‖2,n tends to 0 in probability.Using the definitions of ηn and un , we have

‖un − u‖C(0,T ;L2(I )) � maxt∈[0,T ] ‖ηn(t)‖2,n + ‖vn − u‖C(0,T ;L2(I )). (4.41)

Using Lemma 4.6 and (4.40), we show that ‖un − u‖C(0,T ;L2(I )) tends to 0 inprobability as n → ∞. �

5. Dynamical Models on W-Small-World Graphs

The method developed in the previous sections can be used to derive continuumlimits for a large class of dynamical systems on random graphs. As an application,in this section, we consider dynamical systems on SW graphs [34]. The latter arepopular in modeling networks of diverse nature, because they exhibit the combina-tion of properties that are characteristic to both regular and random graphs, just asseen in many real-life systems [34].

The Nonlinear Heat Equation on W -Random Graphs

First, we introduce a convenient generalization of a SW graph. To this end, letXn be a set of n points from I as defined in (4.2) and let W ∈ W0 be a {0, 1}−valuedgraphon. We assume that W is almost everywhere continuous on I 2 and its supporthas a positive Lebesgue measure. Next, define

Wp(x, y) = (1 − p)W (x, y) + p(1 − W (x, y)), p ∈ [0, 0.5]. (5.1)

Definition 5.1. Gn = G(Wp, Xn) is called a W -small-world (W-SW) graph.

Remark 5.2. Note that for p = 0.5, Wp becomes the Erdos–Rényi graph G(n, 0.5).

Remark 5.3. Using the random set of points from Xn as in (3.1), one constructs aW-SW graph Gn = Gn(W, Xn) generated by a random set of points.

Remark 5.4. Equation (5.1) implies that in the process of construction of the W-SW graph Gn = G(Wp, Xn), the new random edges to be added to the deterministicgraph G(Wp, Xn) are selected from the complement of the edge set E(G(W, Xn)).

It is easy to modify (5.1) to imitate other possible variants of the SW model.For instance, for fixed q ∈ (0, 1),

A) Wp = (1 − p)W + pq and B) Wp = W + pq, p ∈ [0, 1] (5.2)

match the descriptions of the SW networks in [34] and [23,24] respectively.

Theorem 4.3 shows that the continuous model (2.3) with W := Wp approx-imates the discrete network (2.1) on the W-SW graph Gn,p for large n, that is,Equation (2.3) with W = Wp is the continuum limit of the discrete heat equationon the SW graph. We illustrate this result with the continuum limit for the Kuramotomodel on the SW network [8,35].

Example 5.5. The Kuramoto model of coupled identical phase oscillators on theSW graph Gn,p has the following form (cf. [35])

d

dtuni (t) = ω +

∑

j :(i, j)∈E(Gn,p)

sin(2π(unj − uni )), i ∈ [n], (5.3)

where for fixed n ∈ N and i ∈ [n], uni : R → R/Z is interpreted as the phase ofoscillator i and ω is its intrinsic frequency.

For this example, let

Xn ={

0,1

n,

2

n, . . . ,

n − 1

n

}

and

W (x, y) ={

1, d(x, y) � r,0, otherwise,

where d(x, y) = min{|x − y|, 1 − |x − y|} and parameter r ∈ (0, 1) is fixed.With the above definitions, Gn,p is a W-SW graph. In particular, Gn,0 is the k-

nearest-neighbor graph (k = �rn�) (see Fig. 2a), which was used as the underlying

Georgi S. Medvedev

deterministic graph in [34], and Gn,0.5 is the Erdos–Rényi graph G(n, 0.5) (seeFig. 2c). Thus, the family {Gn,p} interpolates between the k-nearest-neighbor graphand the Erdos–Rényi graph. Furthermore, had we chosen to use (5.2A) instead of(5.1), we would have obtained a family of random graph that differs from theoriginal Watts–Strogatz SW model [34] only in minor details.

Theorem 4.3 justifies the following continuum limit for the Kuramoto modelon the W-SW graph

∂

∂tu(x, t) = ω +

∫

IWp(x, y) sin(2π(u(y, t) − u(x, t))) dy. (5.4)

Equation (5.4) can be used to study the stability of q−twisted states, a family ofsteady state solutions of (5.3), just as was done for the k-nearest-neighbor couplednetworks in [8,35]. The analysis of this problem is beyond the scope of this paperand will be presented elsewhere [20].

6. Discussion

Coupled dynamical systems on graphs arise in modeling diverse phenomenain physics, biology, and technology [6,11,16,19,22,30,32]. The dynamics of thesemodels is shaped by the properties of the local dynamical systems at the nodes ofthe graph and the patterns of connections between them. The principal challengeof the mathematical theory of dynamical networks is to elucidate the contributionof the structural properties of the networks to their dynamics. Thus, it is impor-tant to develop analytical techniques, which apply to large classes of networks andreveal the interplay between the local dynamics and network topology. For non-locally coupled dynamical systems, an important (albeit often formal) approachto the analysis of network dynamics has been replacing a discrete model on alarge graph with a continuum (thermodynamic) limit. For networks with nonlineardiffusive coupling the continuum limit is an evolution equation with a nonlocal inte-gral operator modeling nonlinear diffusion. This approach has proved very usefulfor the analysis of nonlocally coupled dynamical systems on deterministic graphs[1,8,14,35].

In applications, one often encounters dynamical networks on random graphs.They are especially important in biology. For example, random graphs are fre-quently used in computational modeling of neuronal systems, because randomconnectivity is often consistent with experimental data. For dynamical networks onrandom graphs, such as SW graphs, even formal continuum limit is not obvious. Onthe other hand, the theory of graph limits provides many examples of convergentsequences of random graphs with relatively simple deterministic limits [4,17,18].In [21], we used the ideas of the theory of graph limits to provide a rigorous mathe-matical justification for taking the continuum limit in a large class of deterministicnetworks. In this paper, we have shown how to derive the limiting equations fordynamical networks on random graphs. Specifically, we studied coupled dynamicalsystems on convergent families of W -random graphs [17,18]. The latter providea convenient analytical framework for modeling random graphs, which includemany important examples arising in applications, such as Erdos–Rényi and SW

The Nonlinear Heat Equation on W -Random Graphs

graphs. We have proven that the solutions of the IVPs for discrete models convergein C(0, T ; L2[0, 1]) norm to their continuous counterpart as the graph size goes toinfinity.

We studied networks for two variants of W -random graphs: those generated bythe random and deterministic sequences respectively. For the discrete problems ofthe first type, the O(n−1/2) convergence is shown. The rate of convergence in thiscase is determined solely by the CLT and holds for all graphons W ∈ W0. Theproof of convergence of the discrete problems of the second type, in addition to theCLT, involves the analysis of the auxiliary IVPs (4.18), (4.19) [see (4.41), (4.35),(4.40), and Lemma 4.6]. The convergence rate of the auxiliary problems dependson the regularity of the graphon W . For instance, Theorem 4.1 in [21] shows thatfor a {0, 1}-valued graphon W , the convergence rate depends on the box-countingdimension of the boundary of the support of W , and may be very slow if the latteris close to 2. Consequently, discrete problems on W -random graphs generated bydeterministic sequences may exhibit slower convergence compared to that of theircounterparts on W -random graphs generated by random sequences. However, theformer are convenient in applications, as they often can be readily related to theexisting random graph models. For example, the classical Watts–Strogatz SW graph[34] can be interpreted as a W -random graph generated by a deterministic sequence.In Section 5, we used this fact to drive the continuum limit for dynamical systems onSW networks as an illustration of our method. We believe that the continuum limitanalyzed in this paper will become a useful tool for studying coupled dynamicalsystems on random graphs.

Acknowledgements. This work was supported in part by the NSF Grant DMS 1109367.

References

1. Abrams D.M., Strogatz, S.H.: Chimera states in a ring of nonlocally coupled oscil-lators. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 16(1), 21–37 (2006)

2. Billingsley, P.: Probability and Measure. Willey, New York, 19953. Bollobas, B.: Random Graph. Cambridge University Press, Cambridge, 20014. Borgs, C., Chayes, J., Lovász, L., Sós, V., Vesztergombi, K.: Limits of randomly

grown graph sequences. Eur. J. Comb. 32, 985–999 (2011)5. Borgs, C., Chayes, J.T., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent

sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv.Math. 219(6), 1801–1851 (2008)

6. Dorfler, F., Bullo, F.,: Synchronization and transient stability in power networks andnon-uniform Kuramoto oscillators. SICON 50(3), 1616–1642 (2012)

7. Ermentrout, G.B., Kopell, N.: Multiple pulse interactions and averaging in systemsof coupled neural oscillators. J. Math. Biol. 29, 195–217 (1991)

8. Girnyk, T., Hasler, M., Maistrenko, Y.: Multistability of twisted states in non-locallycoupled Kuramoto-type models. Chaos 22, 013114 (2012)

9. Hoppensteadt, F.C., Izhikevich, E.M.: Weakly Connected Neural Networks. Springer,Berlin, 1997

10. Janson, S., Luczak, T., Rucinski, A.: Random Graphs. Wiley, Chichester, 201111. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer, Berlin, 198412. Kuramoto, Y.: Cooperative dynamics of oscillator community. Prog. Theor. Phys.

Suppl. 79, 223–240 (1984)

Georgi S. Medvedev

13. Kuramoto, Y.: Scaling behavior of turbulent oscillators with nonlocal interaction. Prog.Theor. Phys. 94, 321–330 (1995)

14. Kuramoto, Y., Battogtokh, D.: Coexistence of coherence and incoherence in nonlo-cally coupled phase oscillators. Nonlinear Phenom. Complex Syst. 5, 380–385 (2002)

15. Laing, C.R.: Chimera states in heterogeneous networks. Chaos 19, 013113 (2009)16. Li, R.D., Erneux, T.: Preferential instability in arrays of coupled lasers. Phys. Rev. A

46, 4252–4260 (1992)17. Lovász, L.: Large Networks and Graph Limits. American Mathematical Society, Prov-

idence, 201218. Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Combin. Theory Ser. B

96(6), 933–957 (2006)19. Medvedev, G.S.: Stochastic stability of continuous time consensus protocols. SIAM J.

Control Optim. 50(4), 1859–1885 (2012)20. Medvedev, G.S.: Small-world networks of Kuramoto oscillators. ArXiv e-prints (2013)21. Medvedev, G.S.: The nonlinear heat equation on dense graphs and graph limits. ArXiv

e-prints (2013)22. Medvedev, G.S., Zhuravytska, S.: The geometry of spontaneous spiking in neuronal

networks. J. Nonlinear Sci. 22, 689–725 (2012)23. Monasson, R.: Diffusion, localization, and dispersion relations on ‘small-world’ lat-

tices. Eur. Phys. J. B 12, 555–567 (1999)24. Newman, N.E.J., Watts, D.J.: Renormalization group analysis of the small-world net-

work model. Phys. Lett. A 263, 341–346 (1999)25. Omelchenko, I., Hovel, P., Maistrenko, Y., Scholl, E.: Loss of coherence in dynam-

ical networks: spatial chaos and chimera states. Phys. Rev. Lett. 106, 234102 (2011)26. Omelchenko, I., Riemenschneider, B., Hövel, P., Maistrenko, Y., Schöll, E.:

Transition from spatial coherence to incoherence in coupled chaotic systems. Phys.Rev. E 85, 026212 (2012)

27. Omel’chenko, O.E., Maistrenko, Y.L., Tass, P.A.: Chimera states: the natural linkbetween coherence and incoherence. Phys. Rev. Lett. 100, 044105 (2008)

28. Omelchenko, O.E., Wolfrum, M., Maistrenko, Y.: Chimera states as chaotic spa-tiotemporal patterns. Phys. Rev. E 81, 065201 (2010)

29. Ott, E., Antonsen, T.M.: Low dimensional behavior of large systems of globallycoupled oscillators. Chaos 18, 037113 (2008)

30. Phillips, J.R., van der Zant, H.S.J., White, J., Orlando, T.P.: Influence of inducedmagnetic fields on the static properties of Josephson-junction arrays. Phys. Rev. B 47,5219–5229 (1993)

31. Shima, S., Kuramoto, Y.: Rotating spiral waves with phase-randomized core in non-locally coupled oscillators. Phys. Rev. E 69(3), 036213 (2004)

32. Strogatz, S.: Sync. How order emerges from chaos in the universe, nature, and dailylife. Hyperion Books, New York (2003)

33. Tanaka, D., Kuramoto, Y.: Complex Ginzburg–Landau equation with nonlocal cou-pling. Phys. Rev. E 68, 026219 (2003)

34. Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature393, 440–442 (1998)

35. Wiley, D.A., Strogatz, S.H., Girvan, M.: The size of the sync basin. Chaos 16(1),015103, 8 (2006)

The Nonlinear Heat Equation on W -Random Graphs

Department of Mathematics,Drexel University,

3141 Chestnut Street,Philadelphia,

PA 19104,USA.

e-mail: medvedev@drexel.edu

(Received May 9, 2013 / Accepted November 28, 2013)© Springer-Verlag Berlin Heidelberg (2013)